You are able to use your iClicker remote in multiple classes as long as your instructor allows the use of your particular iClicker. Check with your instructor regarding instructions for registering your iClicker remote. Below are two scenarios in which you'd need to buy a new iClicker remote for one of your classes.
You have an original iClicker or iClicker + but your instructor requires iClicker 2.
The iClicker 2 remote has added functionality that allows you to answer numeric or short answer questions and to participate in quizzes. If your instructor will be asking these types of questions you'll need an iClicker 2 to fully participate. This will normally be indicated in the syllabus.
You purchased Reef, but your instructor requires a physical iClicker remote.
Some instructors only allow the use of iClickers to vote in their class. Your instructor will usually specify in the syllabus which iClicker remote is required. Check there first before sending them an email. If you know you'll need to use iClicker in multiple classes, the iClicker 2 is the safest bet, because it will work no matter which types of questions your instructor asks.